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This article is cited in 5 scientific papers (total in 5 papers)
MATHEMATICS
On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $ \mathbb{R}^2$ for the Helmholtz equation
A. S. Il'inskiia, I. S. Polyanskiib, D. E. Stepanovb a Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, GSP-1, Leninskie Gory, Moscow, 119991, Russia
b The Academy of Federal Security Guard Service of the Russian
Federation, ul. Priborostroitel'naya, 35, Orel, 302034, Russia
Abstract:
The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.
Keywords:
internal Dirichlet and Neumann problems, Helmholtz equation, arbitrary polygon, barycentric method, Galerkin method, barycentric coordinates, convergence estimation.
Received: 24.06.2020
Citation:
A. S. Il'inskii, I. S. Polyanskii, D. E. Stepanov, “On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $ \mathbb{R}^2$ for the Helmholtz equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:1 (2021), 3–18
Linking options:
https://www.mathnet.ru/eng/vuu751 https://www.mathnet.ru/eng/vuu/v31/i1/p3
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